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2010-01-30 Aperiodic tilings

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January 30th 2010

Aperiodic tilings

Most of the man-made tilings are periodic. "Periodic" means that there are two linearly independent translations which leaves the tiling invariant. On the picture below, the translations along the edges of the red cell leave the tiling invariant, which is therefore periodic. It is of course very easy to draw periodic tilings: simply decompose the basic cell into tiles, making sure that their edges match at the boundaries of the cell.

It is a bit more difficult to create aperiodic tilings with a limited number of non-isomorphic tiles. I'd like to review here a few techniques I used in (old) works. You can find them in the set "Aperiodic tilings".

20000701

20000701 uses a regular tiling by squares, whose edges have been replaced by a broken line. The orientation of the latter is chosen randomly for each edge, what produces an aperiodic tiling using 38 different tiles. Scroll down on the page of the work for more details.

20000721

20000721 displays a spiral tiling using a single type of tiles. Such spiral tilings exist when the tile is an isocele triangle whose "asymmetric" angle (the one which is not equal to the two others...) divides 180°. To construct it, start from the obvious "radial tiling" displayed below.

20000721

Remark that this tiling contains lines passing through the center of the tiling, so let us shift one of the half-plane bounded by such a line by a multiple of the length of the edge of the tiles...

20000721

... and here is a spiral tiling! The tiling of 20000721 is constructed by twisting a little bit the edges of the tiles of the tiling above. A more interesting spiral tiling appears on the cover (and inside, with explanations) of Grünbaum and Shephard "Tiling and pattern", THE book for anyone interested in tilings.

Finally, I would like to speak about substitution tilings. Consider two tiles, a trapezoid and a rhombus with 60° and 120° angles. The pictures below show that they can both be decomposed into versions of the same tiles rescaled by a factor 2.

Suppose now that you start from a single tile, decompose it according to the rules above and scale it by a factor two. After interating this procedure a few times, the tiling will cover a large portion of the plane. It can be shown rigorously that this way of extending the tiling can be used to cover the whole plane (see again Grünbaum and Shephard). The resulting tiling will in general not be periodic. Here is an example (which was the logo of p-gallery.net, my previous website.

20000903

The Penrose tilings, maybe the most famous aperiodic tilings, can be constructed in this way as well (see the "deflation" section of the wikipedia article). In this case, the deflation is a little bit more complex, in the sense that the decomposition of the tiles into smaller tiles also involves half-tiles.

20001223 displays a part of a "Kite and Dart" Penrose tiling. 20010128, 20010716, 20050423 and 20050424 all display aperiodic substitution tilings.